Extending Quantum Mechanics into the Complex Domain

In this talk I will show that one can replace the axiom in quantum mechanics that the Hamiltonian must be Hermitian by the alternative requirement that the Hamiltonian be PT-symmetric (that is, invariant under combined space and time reflection). I will begin by reviewing the
background and motivation and I will illustrate by using an explicit 2x2
matrix Hamiltonian. I will show how to calculate the spectral zeta
function exactly for a class of PT-symmetric Hamiltonians, and I will
verify the result numerically by using a 10th-order WKB calculation.
Using matched asymptotic expansions, I will show how to calculate
the 1-point correlation function for a $-x^4$ theory. I will then
demonstrate a fourth-order perturbative calculation of the C operator
that is needed to define the inner product in PT-symmetric quantum
mechanics. I will show that this C operator is a Lorentz scalar. (By
contrast, the parity operator is not a Lorentz scalar, and instead it
transforms as a direct product of tensorial representations. This direct
sum is deeply related to the Wilson polynomials.) Finally, I will apply
the notions of PT-symmetric quantum mechanics to a well known
quantum-field-theoretic model called the Lee model. I will give a
simple physical interpretation for the ghost states, which have been
known and misinterpreted for the last 50 years.